Reverse Hardy-Littlewood-Sobolev inequalities

TL;DR

This paper introduces a new family of reverse Hardy-Littlewood-Sobolev inequalities with positive power law kernels, exploring parameter ranges, optimal functions, and the potential for concentration phenomena linked to nonlinear diffusion equations.

Contribution

It presents novel reverse inequalities involving positive power law kernels, characterizes optimal functions, and discusses open questions on concentration effects and their relation to nonlinear diffusion.

Findings

Characterization of admissible parameter ranges.

Identification of optimal functions for the inequalities.

Analysis of concentration phenomena and their connection to diffusion equations.

Abstract

This paper is devoted to a new family of reverse Hardy-Littlewood-Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and characterize the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with nonlinear diffusion equations involving mean field drifts.